Solving fuzzy linear systems using a block representation of generalized inverses: The group inverse

We present an efficient method for solving a singular, n×n fuzzy linear system (FLS), AX˜=Y˜, where the coefficient matrix A is a real matrix, singular or non-singular, using the block structure of the group inverse or any {1}-inverse. A characterization of the block structure of {1}-inverses, in particular, the group inverse, but also the Drazin inverse of the matrix associated to a square FLS is given. Based on the presented necessary and sufficient condition for the existence of a solution, the general solution of a square FLS is obtained. Finally, infinitely many solutions of a singular FLS are presented through many interesting examples.